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<h2>2D Transformations</h2>
<p><a href="#section_list">List of Operators ↓</a></p>
<p>To specify a location in an image, we need a convention how to do so.
Such a convention is set via a coordinate system.
There are different coordinate systems used in HALCON. Here, we explain
the ones used in 2D.
</p>
<p>Pixels are discrete and to address them, we have a coordinate system using
only integer values, the pixel coordinate system.
For a higher accuracy that goes beyond the pixel grid, we need floating
point coordinates, like e.g., <span title="1" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="3.88968em" height="1.12127em" viewBox="0 0 62.234940 17.940399">
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This leads to subpixel accurate coordinate systems.
In HALCON, we have three different implementations of subpixel coordinate
systems:
</p>
<ul>
<li>
<p> Pixel Centered Coordinates, the HALCON Standard Subpixel
Coordinate System
</p>
</li>
<li>
<p> Edge Centered Coordinates
</p>
</li>
<li>
<p> Polar Coordinates
</p>
</li>
</ul>
<p>
Thereof the first two vary only in the coordinate origin,
as visible in the figures below.
Calibration makes it possible to map the image coordinates distances to
real-world distances. For more information about these Calibrated
Coordinates we refer to the <code>“Solution Guide III-C - 3D Vision”</code>.
</p>
<h3>HALCON Standard Coordinate System</h3>
<dl class="generic">

<dt><b><b>Pixel Accurate Coordinate System</b></b></dt>
<dd>
<p>

The pixel coordinate system treats the image as a grid of discrete
elements, the pixels.
In HALCON, we put the origin <span title="2" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="2.33993em" height="1.12127em" viewBox="0 0 37.438858 17.940399">
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</svg></span> in the middle of the
upper left pixel.
Now, we assign the pixel coordinates specifying its row and column like
in a matrix.
</p>
<p>Note that this implies for an image of size height <span title="3" style="vertical-align:-0.165107em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="0.900873em" height="0.789172em" viewBox="0 0 14.413971 12.626755">
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</svg></span> width =
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,0.484955 9.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,18.022964 9.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,33.963181 9.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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</svg></span> pixels that the row coordinate
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</svg></span>, as visualized in the figure
below.
</p>
</dd>

<dt><b><b>Subpixel Accurate Coordinate System: Pixel Centered</b></b></dt>
<dd>


<p>The origin of this coordinate system is in the center of the upper left
image pixel, the axes are in row (r) and column (c) direction, respectively.
Therewith this convention embeds the pixel coordinate system.
The upper left image corner has the coordinates
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</svg></span>.
This convention is called the standard coordinate system, or
also Image Coordinate System.
</p>
<div style="text-align:center;" class="figure">
<table style="margin-left:auto;margin-right:auto">
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</tr>
<tr>
<td align="center">
        (
      1)
    </td>
<td align="center">
        (
      2)
    </td>
</tr>
</table>
<div style="margin-bottom:30px;text-align:center;" class="caption">
Visualization of the HALCON standard pixel and subpixel Cartesian
coordinate systems.
The cross indicates the pixel in the bottom right image corner. Its
center has the coordinates <span title="18" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="2.33993em" height="1.12127em" viewBox="0 0 37.438858 17.940399">
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</svg></span> (in pixel coordinates (1)),
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    <use xlink:href="#GLYPHcmr10_54" x="3.888963"></use>
  </g>
  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,15.067703 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,19.495560 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,27.465652 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,42.520447 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,46.948303 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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</svg></span> (in standard subpixel coordinates (2)).
The circle center has the coordinates <span title="20" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="3.88968em" height="1.12127em" viewBox="0 0 62.234940 17.940399">
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</svg></span>.
</div>
</div>

</dd>
</dl>
<h3>HALCON Non-Standard Cartesian Coordinate System</h3>
<p>If we rotate an image around its origin by <span title="21" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="1.72591em" height="1.12127em" viewBox="0 0 27.614502 17.940399">
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(=90 degrees), we want the two images with touching edges but not
overlapping with each other. Also, scaling the image is not expected to
result in negative image coordinates.
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,27.465652 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,34.550186 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
    <use xlink:href="#GLYPHcmr10_48"></use>
  </g>
  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,42.520447 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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  </g>
  <g transform="matrix(1.600006,-0.000000,-0.000000,1.600006,46.948303 11.985046)" style="fill: rgb(0.000000%,0.000000%,0.000000%);">
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</svg></span> has to be set in an image
corner. This motivates the following coordinate system.
</p>
<dl class="generic">

<dt><b><b>Subpixel Accurate Coordinate System: Edge Centered</b></b></dt>
<dd>
<p>

For this coordinate system we set the origin in the upper left image
corner. Thus the center of the upper left pixel has the coordinates
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</svg><div style="margin-bottom:30px;text-align:center;" class="caption">
Visualization of the HALCON non-standard subpixel Cartesian coordinate
system.
The cross indicates the pixel in the bottom right image corner. Its
center has the coordinates <span title="32" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="3.88968em" height="1.12127em" viewBox="0 0 62.234940 17.940399">
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</svg></span>.
The circle center has the coordinates <span title="33" style="vertical-align:-0.37221em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="3.88968em" height="1.12127em" viewBox="0 0 62.234940 17.940399">
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</div>

<p>For this coordinate system rotations are defined in the mathematically
positive direction and thus counterclockwise.
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Accordingly, the axes have the assignment row: x coordinate, column:
y coordinate.
</p>
<div style="text-align:center;" class="figure">
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the edge centered coordinate system.
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</div>

</dd>
</dl>
<h3>Operators Expecting Parameters in any Cartesian Coordinate System</h3>
<p>该算子 <a href="affine_trans_point_2d.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_point_2d</code></span><span data-if="c" style="display:none"><code>affine_trans_point_2d</code></span><span data-if="cpp" style="display:none"><code>AffineTransPoint2d</code></span><span data-if="com" style="display:none"><code>AffineTransPoint2d</code></span><span data-if="dotnet" style="display:none"><code>AffineTransPoint2d</code></span><span data-if="python" style="display:none"><code>affine_trans_point_2d</code></span></code></a> applies the transformation
given by <i><code><span data-if="hdevelop" style="display:inline"><code>HomMat2D</code></span><span data-if="c" style="display:none"><code>HomMat2D</code></span><span data-if="cpp" style="display:none"><code>HomMat2D</code></span><span data-if="com" style="display:none"><code>HomMat2D</code></span><span data-if="dotnet" style="display:none"><code>homMat2D</code></span><span data-if="python" style="display:none"><code>hom_mat_2d</code></span></code></i> to the point coordinates.
This means, <a href="affine_trans_point_2d.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_point_2d</code></span><span data-if="c" style="display:none"><code>affine_trans_point_2d</code></span><span data-if="cpp" style="display:none"><code>AffineTransPoint2d</code></span><span data-if="com" style="display:none"><code>AffineTransPoint2d</code></span><span data-if="dotnet" style="display:none"><code>AffineTransPoint2d</code></span><span data-if="python" style="display:none"><code>affine_trans_point_2d</code></span></code></a> works
in both Cartesian Coordinate systems, as long you make sure that the point
and the transformation are given in the same coordinate system.
</p>
<p>该算子s <a href="angle_ll.html"><code><span data-if="hdevelop" style="display:inline"><code>angle_ll</code></span><span data-if="c" style="display:none"><code>angle_ll</code></span><span data-if="cpp" style="display:none"><code>AngleLl</code></span><span data-if="com" style="display:none"><code>AngleLl</code></span><span data-if="dotnet" style="display:none"><code>AngleLl</code></span><span data-if="python" style="display:none"><code>angle_ll</code></span></code></a> and <a href="angle_lx.html"><code><span data-if="hdevelop" style="display:inline"><code>angle_lx</code></span><span data-if="c" style="display:none"><code>angle_lx</code></span><span data-if="cpp" style="display:none"><code>AngleLx</code></span><span data-if="com" style="display:none"><code>AngleLx</code></span><span data-if="dotnet" style="display:none"><code>AngleLx</code></span><span data-if="python" style="display:none"><code>angle_lx</code></span></code></a> may take the input
points in pixel centered coordinates, but the returned angle is in the
convention of rotations in a mathematically positive direction, thus
counterclockwise, and with the horizontal axis as <i>0</i>, like in the
edge centered coordinate system.
</p>
<h3>Operators Expecting Parameters in Different Coordinate Systems</h3>
<p>
In HALCON there is also the case that an operator expects its input in
different coordinate systems.
On the one hand, the object is expected in its usual coordinates, the
standard coordinates.
On the other hand, for the transformation matrix <i><code><span data-if="hdevelop" style="display:inline"><code>HomMat2D</code></span><span data-if="c" style="display:none"><code>HomMat2D</code></span><span data-if="cpp" style="display:none"><code>HomMat2D</code></span><span data-if="com" style="display:none"><code>HomMat2D</code></span><span data-if="dotnet" style="display:none"><code>homMat2D</code></span><span data-if="python" style="display:none"><code>hom_mat_2d</code></span></code></i>, the
operator expects edge centered coordinates with their advantages regarding
transformations described above.
该算子 converts the coordinates of the object from HALCON's
standard coordinate system (with the origin in the center of the
upper left pixel) to the edge centered coordinate system
(with the origin in the upper left corner of the upper left pixel).
After the transformation with <i><code><span data-if="hdevelop" style="display:inline"><code>HomMat2D</code></span><span data-if="c" style="display:none"><code>HomMat2D</code></span><span data-if="cpp" style="display:none"><code>HomMat2D</code></span><span data-if="com" style="display:none"><code>HomMat2D</code></span><span data-if="dotnet" style="display:none"><code>homMat2D</code></span><span data-if="python" style="display:none"><code>hom_mat_2d</code></span></code></i>, the result is
converted back to the standard coordinate system.
</p>
<p>These operators are
</p>
<ul>
<li>
<p> <a href="affine_trans_contour_xld.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_contour_xld</code></span><span data-if="c" style="display:none"><code>affine_trans_contour_xld</code></span><span data-if="cpp" style="display:none"><code>AffineTransContourXld</code></span><span data-if="com" style="display:none"><code>AffineTransContourXld</code></span><span data-if="dotnet" style="display:none"><code>AffineTransContourXld</code></span><span data-if="python" style="display:none"><code>affine_trans_contour_xld</code></span></code></a>
</p>
</li>
<li>
<p> <a href="affine_trans_image.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_image</code></span><span data-if="c" style="display:none"><code>affine_trans_image</code></span><span data-if="cpp" style="display:none"><code>AffineTransImage</code></span><span data-if="com" style="display:none"><code>AffineTransImage</code></span><span data-if="dotnet" style="display:none"><code>AffineTransImage</code></span><span data-if="python" style="display:none"><code>affine_trans_image</code></span></code></a>
</p>
</li>
<li>
<p> <a href="affine_trans_image_size.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_image_size</code></span><span data-if="c" style="display:none"><code>affine_trans_image_size</code></span><span data-if="cpp" style="display:none"><code>AffineTransImageSize</code></span><span data-if="com" style="display:none"><code>AffineTransImageSize</code></span><span data-if="dotnet" style="display:none"><code>AffineTransImageSize</code></span><span data-if="python" style="display:none"><code>affine_trans_image_size</code></span></code></a>
</p>
</li>
<li>
<p> <a href="affine_trans_pixel.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_pixel</code></span><span data-if="c" style="display:none"><code>affine_trans_pixel</code></span><span data-if="cpp" style="display:none"><code>AffineTransPixel</code></span><span data-if="com" style="display:none"><code>AffineTransPixel</code></span><span data-if="dotnet" style="display:none"><code>AffineTransPixel</code></span><span data-if="python" style="display:none"><code>affine_trans_pixel</code></span></code></a>
</p>
</li>
<li>
<p> <a href="affine_trans_polygon_xld.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_polygon_xld</code></span><span data-if="c" style="display:none"><code>affine_trans_polygon_xld</code></span><span data-if="cpp" style="display:none"><code>AffineTransPolygonXld</code></span><span data-if="com" style="display:none"><code>AffineTransPolygonXld</code></span><span data-if="dotnet" style="display:none"><code>AffineTransPolygonXld</code></span><span data-if="python" style="display:none"><code>affine_trans_polygon_xld</code></span></code></a>
</p>
</li>
<li>
<p> <a href="affine_trans_region.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_region</code></span><span data-if="c" style="display:none"><code>affine_trans_region</code></span><span data-if="cpp" style="display:none"><code>AffineTransRegion</code></span><span data-if="com" style="display:none"><code>AffineTransRegion</code></span><span data-if="dotnet" style="display:none"><code>AffineTransRegion</code></span><span data-if="python" style="display:none"><code>affine_trans_region</code></span></code></a>
</p>
</li>
<li>
<p> <a href="projective_trans_contour_xld.html"><code><span data-if="hdevelop" style="display:inline"><code>projective_trans_contour_xld</code></span><span data-if="c" style="display:none"><code>projective_trans_contour_xld</code></span><span data-if="cpp" style="display:none"><code>ProjectiveTransContourXld</code></span><span data-if="com" style="display:none"><code>ProjectiveTransContourXld</code></span><span data-if="dotnet" style="display:none"><code>ProjectiveTransContourXld</code></span><span data-if="python" style="display:none"><code>projective_trans_contour_xld</code></span></code></a>
</p>
</li>
<li>
<p> <a href="projective_trans_image.html"><code><span data-if="hdevelop" style="display:inline"><code>projective_trans_image</code></span><span data-if="c" style="display:none"><code>projective_trans_image</code></span><span data-if="cpp" style="display:none"><code>ProjectiveTransImage</code></span><span data-if="com" style="display:none"><code>ProjectiveTransImage</code></span><span data-if="dotnet" style="display:none"><code>ProjectiveTransImage</code></span><span data-if="python" style="display:none"><code>projective_trans_image</code></span></code></a>
</p>
</li>
<li>
<p> <a href="projective_trans_image_size.html"><code><span data-if="hdevelop" style="display:inline"><code>projective_trans_image_size</code></span><span data-if="c" style="display:none"><code>projective_trans_image_size</code></span><span data-if="cpp" style="display:none"><code>ProjectiveTransImageSize</code></span><span data-if="com" style="display:none"><code>ProjectiveTransImageSize</code></span><span data-if="dotnet" style="display:none"><code>ProjectiveTransImageSize</code></span><span data-if="python" style="display:none"><code>projective_trans_image_size</code></span></code></a>
</p>
</li>
<li>
<p> <a href="projective_trans_pixel.html"><code><span data-if="hdevelop" style="display:inline"><code>projective_trans_pixel</code></span><span data-if="c" style="display:none"><code>projective_trans_pixel</code></span><span data-if="cpp" style="display:none"><code>ProjectiveTransPixel</code></span><span data-if="com" style="display:none"><code>ProjectiveTransPixel</code></span><span data-if="dotnet" style="display:none"><code>ProjectiveTransPixel</code></span><span data-if="python" style="display:none"><code>projective_trans_pixel</code></span></code></a>
</p>
</li>
<li>
<p> <a href="projective_trans_region.html"><code><span data-if="hdevelop" style="display:inline"><code>projective_trans_region</code></span><span data-if="c" style="display:none"><code>projective_trans_region</code></span><span data-if="cpp" style="display:none"><code>ProjectiveTransRegion</code></span><span data-if="com" style="display:none"><code>ProjectiveTransRegion</code></span><span data-if="dotnet" style="display:none"><code>ProjectiveTransRegion</code></span><span data-if="python" style="display:none"><code>projective_trans_region</code></span></code></a>
</p>
</li>
</ul>
<p>A matrix representing a transformation in pixel centered coordinates can
be converted to represent the same transformation (e.g., a rotation around
the same point) written in edge centered coordinates, e.g., through
</p>
<div style="text-align:center"><table style="width:90%;margin-left:auto;margin-right:auto" class="layout">
<col span="1" style="width:90%;">
<tr><td style="text-align:left">
<a href="hom_mat2d_translate.html"><code><span data-if="hdevelop" style="display:inline"><code>hom_mat2d_translate(HomMat2D, 0.5, 0.5, HomMat2DTmp)</code></span><span data-if="c" style="display:none"><code>hom_mat2d_translate(HomMat2D, 0.5, 0.5, HomMat2DTmp)</code></span><span data-if="cpp" style="display:none"><code>HomMat2dTranslate(HomMat2D, 0.5, 0.5, HomMat2DTmp)</code></span><span data-if="com" style="display:none"><code>HomMat2dTranslate(HomMat2D, 0.5, 0.5, HomMat2DTmp)</code></span><span data-if="dotnet" style="display:none"><code>HomMat2dTranslate(HomMat2D, 0.5, 0.5, HomMat2DTmp)</code></span><span data-if="python" style="display:none"><code>hom_mat2d_translate(HomMat2D, 0.5, 0.5, HomMat2DTmp)</code></span></code></a>
</td></tr>
<tr><td style="text-align:left">
<a href="hom_mat2d_translate_local.html"><code><span data-if="hdevelop" style="display:inline"><code>hom_mat2d_translate_local(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)</code></span><span data-if="c" style="display:none"><code>hom_mat2d_translate_local(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)</code></span><span data-if="cpp" style="display:none"><code>HomMat2dTranslateLocal(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)</code></span><span data-if="com" style="display:none"><code>HomMat2dTranslateLocal(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)</code></span><span data-if="dotnet" style="display:none"><code>HomMat2dTranslateLocal(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)</code></span><span data-if="python" style="display:none"><code>hom_mat2d_translate_local(HomMat2DTmp, -0.5, -0.5, HomMat2DAdapted)</code></span></code></a>
</td></tr>
</table></div>
<p>Note, 该算子s beginning with <code>projective_</code> mentioned above
use a projective transformation matrix. These transformation matrices can,
e.g., be obtained from a 3D camera pose. Doing so, the matrix used is
written in a projection of the xy-plane within the 3D coordinate system.
Accordingly, the axes have the assignment row: y coordinate, column:
x coordinate and therewith the coordinates need to be converted.
</p>
<h3>Shape-based Matching and Correlation-based Matching</h3>
<p>Results from shape-based matching, like e.g.,
<a href="find_generic_shape_model.html"><code><span data-if="hdevelop" style="display:inline"><code>find_generic_shape_model</code></span><span data-if="c" style="display:none"><code>find_generic_shape_model</code></span><span data-if="cpp" style="display:none"><code>FindGenericShapeModel</code></span><span data-if="com" style="display:none"><code>FindGenericShapeModel</code></span><span data-if="dotnet" style="display:none"><code>FindGenericShapeModel</code></span><span data-if="python" style="display:none"><code>find_generic_shape_model</code></span></code></a>, are given in edge centered coordinates.
The returned matches are already transformed. The respective homographic
transformation matrices can be retrieved using
<a href="get_generic_shape_model_result.html"><code><span data-if="hdevelop" style="display:inline"><code>get_generic_shape_model_result</code></span><span data-if="c" style="display:none"><code>get_generic_shape_model_result</code></span><span data-if="cpp" style="display:none"><code>GetGenericShapeModelResult</code></span><span data-if="com" style="display:none"><code>GetGenericShapeModelResult</code></span><span data-if="dotnet" style="display:none"><code>GetGenericShapeModelResult</code></span><span data-if="python" style="display:none"><code>get_generic_shape_model_result</code></span></code></a>.
</p>
<p>Results from correlation-based matching, like e.g.,
<a href="find_ncc_model.html"><code><span data-if="hdevelop" style="display:inline"><code>find_ncc_model</code></span><span data-if="c" style="display:none"><code>find_ncc_model</code></span><span data-if="cpp" style="display:none"><code>FindNccModel</code></span><span data-if="com" style="display:none"><code>FindNccModel</code></span><span data-if="dotnet" style="display:none"><code>FindNccModel</code></span><span data-if="python" style="display:none"><code>find_ncc_model</code></span></code></a> and <a href="find_ncc_models.html"><code><span data-if="hdevelop" style="display:inline"><code>find_ncc_models</code></span><span data-if="c" style="display:none"><code>find_ncc_models</code></span><span data-if="cpp" style="display:none"><code>FindNccModels</code></span><span data-if="com" style="display:none"><code>FindNccModels</code></span><span data-if="dotnet" style="display:none"><code>FindNccModels</code></span><span data-if="python" style="display:none"><code>find_ncc_models</code></span></code></a>,
are computed in edge centered coordinates as well, however the parameters
for the transformation are returned separately.
With these results one can create a transformation
<i><code><span data-if="hdevelop" style="display:inline"><code>HomMat2D</code></span><span data-if="c" style="display:none"><code>HomMat2D</code></span><span data-if="cpp" style="display:none"><code>HomMat2D</code></span><span data-if="com" style="display:none"><code>HomMat2D</code></span><span data-if="dotnet" style="display:none"><code>homMat2D</code></span><span data-if="python" style="display:none"><code>hom_mat_2d</code></span></code></i> directly applicable for,
e.g., <a href="affine_trans_contour_xld.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_contour_xld</code></span><span data-if="c" style="display:none"><code>affine_trans_contour_xld</code></span><span data-if="cpp" style="display:none"><code>AffineTransContourXld</code></span><span data-if="com" style="display:none"><code>AffineTransContourXld</code></span><span data-if="dotnet" style="display:none"><code>AffineTransContourXld</code></span><span data-if="python" style="display:none"><code>affine_trans_contour_xld</code></span></code></a>
and the other operators listed in the paragraph above, entitled
<code>Operators Expecting Parameters in Different Coordinate Systems</code>.
</p>
<p>To display the results found by correlation-based matching, we highly
recommend the usage of the procedure
<code>dev_display_ncc_matching_results</code>.
</p>
<p>In the following images we give an example how a displayed
match may look when using the transformation matrix in the correct and
the erroneous coordinate system, respectively.
For the latter one, shown in image (3), the transition matrix is given in
pixel centered coordinates as well and therefore the match shown by
<a href="affine_trans_contour_xld.html"><code><span data-if="hdevelop" style="display:inline"><code>affine_trans_contour_xld</code></span><span data-if="c" style="display:none"><code>affine_trans_contour_xld</code></span><span data-if="cpp" style="display:none"><code>AffineTransContourXld</code></span><span data-if="com" style="display:none"><code>AffineTransContourXld</code></span><span data-if="dotnet" style="display:none"><code>AffineTransContourXld</code></span><span data-if="python" style="display:none"><code>affine_trans_contour_xld</code></span></code></a> is off by 0.5 pixels.
Note, this effect is only visible when a rotation is involved.
</p>
<p><div style="text-align:left|right|center">
<div style="text-align:left|right|center;" class="figure">
<table style="margin-left:auto;margin-right:auto">
<tr>
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</svg></td>
</tr>
<tr>
<td align="center">
        (
      1)
    </td>
<td align="center">
        (
      2)
    </td>
<td align="center">
        (
      3)
    </td>
</tr>
</table>
<div style="margin-bottom:30px;text-align:left|right|center;" class="caption">
The original image of the paperclip (1), a part of the match where the
inputs are given in the correct coordinates (2), and a match with
inputs given in wrong coordinates.
</div>
</div>
</div>
</p>
<h3>Non-Cartesian Coordinate Systems</h3>
<dl class="generic">

<dt><b><b>Subpixel Accurate Coordinate System: Polar Coordinates</b></b></dt>
<dd>
<p>

In polar coordinates, points are defined through a distance and an angle.
The distance is called the radial coordinate and is given in relation to
the fix point, the pole. The angular coordinate is given with respect to a
defined axis, the polar axis.
In HALCON, the pole is specified by
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and the polar axis is the horizontal axis.
The angular coordinate is given in radians.
</p>
<p>After a transformation with <a href="polar_trans_image_ext.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_image_ext</code></span><span data-if="c" style="display:none"><code>polar_trans_image_ext</code></span><span data-if="cpp" style="display:none"><code>PolarTransImageExt</code></span><span data-if="com" style="display:none"><code>PolarTransImageExt</code></span><span data-if="dotnet" style="display:none"><code>PolarTransImageExt</code></span><span data-if="python" style="display:none"><code>polar_trans_image_ext</code></span></code></a>, the
upper left pixel in the output image always corresponds to the
point in the input image that is specified by <i><code><span data-if="hdevelop" style="display:inline"><code>RadiusStart</code></span><span data-if="c" style="display:none"><code>RadiusStart</code></span><span data-if="cpp" style="display:none"><code>RadiusStart</code></span><span data-if="com" style="display:none"><code>RadiusStart</code></span><span data-if="dotnet" style="display:none"><code>radiusStart</code></span><span data-if="python" style="display:none"><code>radius_start</code></span></code></i>
and <i><code><span data-if="hdevelop" style="display:inline"><code>AngleStart</code></span><span data-if="c" style="display:none"><code>AngleStart</code></span><span data-if="cpp" style="display:none"><code>AngleStart</code></span><span data-if="com" style="display:none"><code>AngleStart</code></span><span data-if="dotnet" style="display:none"><code>angleStart</code></span><span data-if="python" style="display:none"><code>angle_start</code></span></code></i>. Analogously, the lower right pixel in the
output image corresponds to the point in the input image that
is specified by <i><code><span data-if="hdevelop" style="display:inline"><code>RadiusEnd</code></span><span data-if="c" style="display:none"><code>RadiusEnd</code></span><span data-if="cpp" style="display:none"><code>RadiusEnd</code></span><span data-if="com" style="display:none"><code>RadiusEnd</code></span><span data-if="dotnet" style="display:none"><code>radiusEnd</code></span><span data-if="python" style="display:none"><code>radius_end</code></span></code></i> and <i><code><span data-if="hdevelop" style="display:inline"><code>AngleEnd</code></span><span data-if="c" style="display:none"><code>AngleEnd</code></span><span data-if="cpp" style="display:none"><code>AngleEnd</code></span><span data-if="com" style="display:none"><code>AngleEnd</code></span><span data-if="dotnet" style="display:none"><code>angleEnd</code></span><span data-if="python" style="display:none"><code>angle_end</code></span></code></i>. In the
usual mode (<i><code><span data-if="hdevelop" style="display:inline"><code>AngleStart</code></span><span data-if="c" style="display:none"><code>AngleStart</code></span><span data-if="cpp" style="display:none"><code>AngleStart</code></span><span data-if="com" style="display:none"><code>AngleStart</code></span><span data-if="dotnet" style="display:none"><code>angleStart</code></span><span data-if="python" style="display:none"><code>angle_start</code></span></code></i> &lt; <i><code><span data-if="hdevelop" style="display:inline"><code>AngleEnd</code></span><span data-if="c" style="display:none"><code>AngleEnd</code></span><span data-if="cpp" style="display:none"><code>AngleEnd</code></span><span data-if="com" style="display:none"><code>AngleEnd</code></span><span data-if="dotnet" style="display:none"><code>angleEnd</code></span><span data-if="python" style="display:none"><code>angle_end</code></span></code></i> and
<i><code><span data-if="hdevelop" style="display:inline"><code>RadiusStart</code></span><span data-if="c" style="display:none"><code>RadiusStart</code></span><span data-if="cpp" style="display:none"><code>RadiusStart</code></span><span data-if="com" style="display:none"><code>RadiusStart</code></span><span data-if="dotnet" style="display:none"><code>radiusStart</code></span><span data-if="python" style="display:none"><code>radius_start</code></span></code></i> &lt; <i><code><span data-if="hdevelop" style="display:inline"><code>RadiusEnd</code></span><span data-if="c" style="display:none"><code>RadiusEnd</code></span><span data-if="cpp" style="display:none"><code>RadiusEnd</code></span><span data-if="com" style="display:none"><code>RadiusEnd</code></span><span data-if="dotnet" style="display:none"><code>radiusEnd</code></span><span data-if="python" style="display:none"><code>radius_end</code></span></code></i>), the polar
transformation is performed in the mathematically positive
orientation (counterclockwise). Furthermore, points with smaller
radius lie in the upper part of the output image. By suitably
exchanging the values of these parameters (e.g., <i><code><span data-if="hdevelop" style="display:inline"><code>AngleStart</code></span><span data-if="c" style="display:none"><code>AngleStart</code></span><span data-if="cpp" style="display:none"><code>AngleStart</code></span><span data-if="com" style="display:none"><code>AngleStart</code></span><span data-if="dotnet" style="display:none"><code>angleStart</code></span><span data-if="python" style="display:none"><code>angle_start</code></span></code></i>
&gt; <i><code><span data-if="hdevelop" style="display:inline"><code>AngleEnd</code></span><span data-if="c" style="display:none"><code>AngleEnd</code></span><span data-if="cpp" style="display:none"><code>AngleEnd</code></span><span data-if="com" style="display:none"><code>AngleEnd</code></span><span data-if="dotnet" style="display:none"><code>angleEnd</code></span><span data-if="python" style="display:none"><code>angle_end</code></span></code></i> or <i><code><span data-if="hdevelop" style="display:inline"><code>RadiusStart</code></span><span data-if="c" style="display:none"><code>RadiusStart</code></span><span data-if="cpp" style="display:none"><code>RadiusStart</code></span><span data-if="com" style="display:none"><code>RadiusStart</code></span><span data-if="dotnet" style="display:none"><code>radiusStart</code></span><span data-if="python" style="display:none"><code>radius_start</code></span></code></i> &gt;
<i><code><span data-if="hdevelop" style="display:inline"><code>RadiusEnd</code></span><span data-if="c" style="display:none"><code>RadiusEnd</code></span><span data-if="cpp" style="display:none"><code>RadiusEnd</code></span><span data-if="com" style="display:none"><code>RadiusEnd</code></span><span data-if="dotnet" style="display:none"><code>radiusEnd</code></span><span data-if="python" style="display:none"><code>radius_end</code></span></code></i>), any desired orientation of the output image can
be achieved.
</p>
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</tr>
<tr>
<td align="center">
        (
      1)
    </td>
<td align="center">
        (
      2)
    </td>
<td align="center">
        (
      3)
    </td>
</tr>
</table>
<div style="margin-bottom:30px;text-align:center;" class="caption">
As an example, we show an annular arc defined by its pole
(<i><code><span data-if="hdevelop" style="display:inline"><code>Row</code></span><span data-if="c" style="display:none"><code>Row</code></span><span data-if="cpp" style="display:none"><code>Row</code></span><span data-if="com" style="display:none"><code>Row</code></span><span data-if="dotnet" style="display:none"><code>row</code></span><span data-if="python" style="display:none"><code>row</code></span></code></i>,<i><code><span data-if="hdevelop" style="display:inline"><code>Column</code></span><span data-if="c" style="display:none"><code>Column</code></span><span data-if="cpp" style="display:none"><code>Column</code></span><span data-if="com" style="display:none"><code>Column</code></span><span data-if="dotnet" style="display:none"><code>column</code></span><span data-if="python" style="display:none"><code>column</code></span></code></i>) (+), the polar axis (...),
two angular coordinates <i><code><span data-if="hdevelop" style="display:inline"><code>AngleStart</code></span><span data-if="c" style="display:none"><code>AngleStart</code></span><span data-if="cpp" style="display:none"><code>AngleStart</code></span><span data-if="com" style="display:none"><code>AngleStart</code></span><span data-if="dotnet" style="display:none"><code>angleStart</code></span><span data-if="python" style="display:none"><code>angle_start</code></span></code></i>
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<i><code><span data-if="hdevelop" style="display:inline"><code>AngleEnd</code></span><span data-if="c" style="display:none"><code>AngleEnd</code></span><span data-if="cpp" style="display:none"><code>AngleEnd</code></span><span data-if="com" style="display:none"><code>AngleEnd</code></span><span data-if="dotnet" style="display:none"><code>angleEnd</code></span><span data-if="python" style="display:none"><code>angle_end</code></span></code></i> (<span title="38" style="vertical-align:-0.0710211em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="1.26513em" height="0.767042em" viewBox="0 0 20.242157 12.272675">
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</svg></span>) and two radial
coordinates <i><code><span data-if="hdevelop" style="display:inline"><code>RadiusStart</code></span><span data-if="c" style="display:none"><code>RadiusStart</code></span><span data-if="cpp" style="display:none"><code>RadiusStart</code></span><span data-if="com" style="display:none"><code>RadiusStart</code></span><span data-if="dotnet" style="display:none"><code>radiusStart</code></span><span data-if="python" style="display:none"><code>radius_start</code></span></code></i> (<span title="39" style="vertical-align:-0.0710211em" class="math"><!-- Created by MetaPost 1.902 on 2023.05.15:2033 --><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="1.10128em" height="0.767042em" viewBox="0 0 17.620544 12.272675">
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</svg></span>), <i><code><span data-if="hdevelop" style="display:inline"><code>RadiusEnd</code></span><span data-if="c" style="display:none"><code>RadiusEnd</code></span><span data-if="cpp" style="display:none"><code>RadiusEnd</code></span><span data-if="com" style="display:none"><code>RadiusEnd</code></span><span data-if="dotnet" style="display:none"><code>radiusEnd</code></span><span data-if="python" style="display:none"><code>radius_end</code></span></code></i>
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</svg></span>).
(1) The original image and the parameters defining the annular arc.
(2) The annular arc, shown in a figure where the polar
coordinates form an equidistant grid obtained by
<a href="polar_trans_image_ext.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_image_ext</code></span><span data-if="c" style="display:none"><code>polar_trans_image_ext</code></span><span data-if="cpp" style="display:none"><code>PolarTransImageExt</code></span><span data-if="com" style="display:none"><code>PolarTransImageExt</code></span><span data-if="dotnet" style="display:none"><code>PolarTransImageExt</code></span><span data-if="python" style="display:none"><code>polar_trans_image_ext</code></span></code></a>.
(3) The annular arc in the representation of the original image.
The Cartesian coordinates have been obtained through
<a href="polar_trans_image_inv.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_image_inv</code></span><span data-if="c" style="display:none"><code>polar_trans_image_inv</code></span><span data-if="cpp" style="display:none"><code>PolarTransImageInv</code></span><span data-if="com" style="display:none"><code>PolarTransImageInv</code></span><span data-if="dotnet" style="display:none"><code>PolarTransImageInv</code></span><span data-if="python" style="display:none"><code>polar_trans_image_inv</code></span></code></a> on image (2).
The origin is in the center of the pixel in the upper left corner.
</div>
</div>

<p>Polar coordinates are used by the following operators:
</p>
<ul>
<li>
<p> <a href="polar_trans_image_ext.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_image_ext</code></span><span data-if="c" style="display:none"><code>polar_trans_image_ext</code></span><span data-if="cpp" style="display:none"><code>PolarTransImageExt</code></span><span data-if="com" style="display:none"><code>PolarTransImageExt</code></span><span data-if="dotnet" style="display:none"><code>PolarTransImageExt</code></span><span data-if="python" style="display:none"><code>polar_trans_image_ext</code></span></code></a>
</p>
</li>
<li>
<p> <a href="polar_trans_image_inv.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_image_inv</code></span><span data-if="c" style="display:none"><code>polar_trans_image_inv</code></span><span data-if="cpp" style="display:none"><code>PolarTransImageInv</code></span><span data-if="com" style="display:none"><code>PolarTransImageInv</code></span><span data-if="dotnet" style="display:none"><code>PolarTransImageInv</code></span><span data-if="python" style="display:none"><code>polar_trans_image_inv</code></span></code></a>
</p>
</li>
<li>
<p> <a href="polar_trans_region.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_region</code></span><span data-if="c" style="display:none"><code>polar_trans_region</code></span><span data-if="cpp" style="display:none"><code>PolarTransRegion</code></span><span data-if="com" style="display:none"><code>PolarTransRegion</code></span><span data-if="dotnet" style="display:none"><code>PolarTransRegion</code></span><span data-if="python" style="display:none"><code>polar_trans_region</code></span></code></a>
</p>
</li>
<li>
<p> <a href="polar_trans_region_inv.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_region_inv</code></span><span data-if="c" style="display:none"><code>polar_trans_region_inv</code></span><span data-if="cpp" style="display:none"><code>PolarTransRegionInv</code></span><span data-if="com" style="display:none"><code>PolarTransRegionInv</code></span><span data-if="dotnet" style="display:none"><code>PolarTransRegionInv</code></span><span data-if="python" style="display:none"><code>polar_trans_region_inv</code></span></code></a>
</p>
</li>
<li>
<p> <a href="polar_trans_contour_xld.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_contour_xld</code></span><span data-if="c" style="display:none"><code>polar_trans_contour_xld</code></span><span data-if="cpp" style="display:none"><code>PolarTransContourXld</code></span><span data-if="com" style="display:none"><code>PolarTransContourXld</code></span><span data-if="dotnet" style="display:none"><code>PolarTransContourXld</code></span><span data-if="python" style="display:none"><code>polar_trans_contour_xld</code></span></code></a>
</p>
</li>
<li>
<p> <a href="polar_trans_contour_xld_inv.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_contour_xld_inv</code></span><span data-if="c" style="display:none"><code>polar_trans_contour_xld_inv</code></span><span data-if="cpp" style="display:none"><code>PolarTransContourXldInv</code></span><span data-if="com" style="display:none"><code>PolarTransContourXldInv</code></span><span data-if="dotnet" style="display:none"><code>PolarTransContourXldInv</code></span><span data-if="python" style="display:none"><code>polar_trans_contour_xld_inv</code></span></code></a>
</p>
</li>
<li>
<p> <a href="polar_trans_image.html"><code><span data-if="hdevelop" style="display:inline"><code>polar_trans_image</code></span><span data-if="c" style="display:none"><code>polar_trans_image</code></span><span data-if="cpp" style="display:none"><code>PolarTransImage</code></span><span data-if="com" style="display:none"><code>PolarTransImage</code></span><span data-if="dotnet" style="display:none"><code>PolarTransImage</code></span><span data-if="python" style="display:none"><code>polar_trans_image</code></span></code></a> (legacy)
</p>
</li>
</ul>

</dd>
</dl>
<h3>Images with a reduced domain, regions, and models</h3>
<p>In the part before we spoke about coordinates of images.
When it comes to the location of the origin of the coordinate system used,
images with reduced domains, regions, and models are treated differently
than images.
</p>
<dl class="generic">

<dt><b><b>Images with a reduced domain and regions</b></b></dt>
<dd>
<p>

Both images with a reduced domain and regions keep the
coordinate system of the image from which they were created.
This means, they inherit the origin and the points keep the coordinate
values they had in the original image.
</p>
</dd>

<dt><b><b>Models</b></b></dt>
<dd><p>

Models, on the other side, can have a local coordinate system.
E.g., models obtained over <a href="create_generic_shape_model.html"><code><span data-if="hdevelop" style="display:inline"><code>create_generic_shape_model</code></span><span data-if="c" style="display:none"><code>create_generic_shape_model</code></span><span data-if="cpp" style="display:none"><code>CreateGenericShapeModel</code></span><span data-if="com" style="display:none"><code>CreateGenericShapeModel</code></span><span data-if="dotnet" style="display:none"><code>CreateGenericShapeModel</code></span><span data-if="python" style="display:none"><code>create_generic_shape_model</code></span></code></a> have their
origin in the center of gravity of the ROI they are created from.
For further information see the
<code>“Solution Guide II-B - Matching”</code>.
</p></dd>
</dl>
<h3>Calibrated Coordinates</h3>
<p>While working with pixel units, we can not extract any information about
real-world distances directly.
When a camera is calibrated, it is possible to rectify the images.
In this case one can assign world coordinates to the image.
For further information we refer to the
<code>“Solution Guide III-C - 3D Vision”</code>.
</p>
<hr>
<h4 id="section_list">算子列表</h4>
<dl>
<dt><a href="affine_trans_pixel.html"><code><span data-if="hdevelop" style="display:inline;">affine_trans_pixel</span><span data-if="dotnet" style="display:none;">AffineTransPixel</span><span data-if="python" style="display:none;">affine_trans_pixel</span><span data-if="cpp" style="display:none;">AffineTransPixel</span><span data-if="c" style="display:none;">affine_trans_pixel</span></code></a></dt>
<dd>Apply an arbitrary affine 2D transformation to pixel coordinates.</dd>
</dl>
<dl>
<dt><a href="affine_trans_point_2d.html"><code><span data-if="hdevelop" style="display:inline;">affine_trans_point_2d</span><span data-if="dotnet" style="display:none;">AffineTransPoint2d</span><span data-if="python" style="display:none;">affine_trans_point_2d</span><span data-if="cpp" style="display:none;">AffineTransPoint2d</span><span data-if="c" style="display:none;">affine_trans_point_2d</span></code></a></dt>
<dd>Apply an arbitrary affine 2D transformation to points.</dd>
</dl>
<dl>
<dt><a href="deserialize_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">deserialize_hom_mat2d</span><span data-if="dotnet" style="display:none;">DeserializeHomMat2d</span><span data-if="python" style="display:none;">deserialize_hom_mat2d</span><span data-if="cpp" style="display:none;">DeserializeHomMat2d</span><span data-if="c" style="display:none;">deserialize_hom_mat2d</span></code></a></dt>
<dd>Deserialize a serialized homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_compose.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_compose</span><span data-if="dotnet" style="display:none;">HomMat2dCompose</span><span data-if="python" style="display:none;">hom_mat2d_compose</span><span data-if="cpp" style="display:none;">HomMat2dCompose</span><span data-if="c" style="display:none;">hom_mat2d_compose</span></code></a></dt>
<dd>Multiply two homogeneous 2D transformation matrices.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_determinant.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_determinant</span><span data-if="dotnet" style="display:none;">HomMat2dDeterminant</span><span data-if="python" style="display:none;">hom_mat2d_determinant</span><span data-if="cpp" style="display:none;">HomMat2dDeterminant</span><span data-if="c" style="display:none;">hom_mat2d_determinant</span></code></a></dt>
<dd>Compute the determinant of a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_identity.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_identity</span><span data-if="dotnet" style="display:none;">HomMat2dIdentity</span><span data-if="python" style="display:none;">hom_mat2d_identity</span><span data-if="cpp" style="display:none;">HomMat2dIdentity</span><span data-if="c" style="display:none;">hom_mat2d_identity</span></code></a></dt>
<dd>Generate the homogeneous transformation matrix of the identical 2D
transformation.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_invert.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_invert</span><span data-if="dotnet" style="display:none;">HomMat2dInvert</span><span data-if="python" style="display:none;">hom_mat2d_invert</span><span data-if="cpp" style="display:none;">HomMat2dInvert</span><span data-if="c" style="display:none;">hom_mat2d_invert</span></code></a></dt>
<dd>Invert a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_reflect.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_reflect</span><span data-if="dotnet" style="display:none;">HomMat2dReflect</span><span data-if="python" style="display:none;">hom_mat2d_reflect</span><span data-if="cpp" style="display:none;">HomMat2dReflect</span><span data-if="c" style="display:none;">hom_mat2d_reflect</span></code></a></dt>
<dd>Add a reflection to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_reflect_local.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_reflect_local</span><span data-if="dotnet" style="display:none;">HomMat2dReflectLocal</span><span data-if="python" style="display:none;">hom_mat2d_reflect_local</span><span data-if="cpp" style="display:none;">HomMat2dReflectLocal</span><span data-if="c" style="display:none;">hom_mat2d_reflect_local</span></code></a></dt>
<dd>Add a reflection to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_rotate.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_rotate</span><span data-if="dotnet" style="display:none;">HomMat2dRotate</span><span data-if="python" style="display:none;">hom_mat2d_rotate</span><span data-if="cpp" style="display:none;">HomMat2dRotate</span><span data-if="c" style="display:none;">hom_mat2d_rotate</span></code></a></dt>
<dd>Add a rotation to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_rotate_local.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_rotate_local</span><span data-if="dotnet" style="display:none;">HomMat2dRotateLocal</span><span data-if="python" style="display:none;">hom_mat2d_rotate_local</span><span data-if="cpp" style="display:none;">HomMat2dRotateLocal</span><span data-if="c" style="display:none;">hom_mat2d_rotate_local</span></code></a></dt>
<dd>Add a rotation to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_scale.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_scale</span><span data-if="dotnet" style="display:none;">HomMat2dScale</span><span data-if="python" style="display:none;">hom_mat2d_scale</span><span data-if="cpp" style="display:none;">HomMat2dScale</span><span data-if="c" style="display:none;">hom_mat2d_scale</span></code></a></dt>
<dd>Add a scaling to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_scale_local.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_scale_local</span><span data-if="dotnet" style="display:none;">HomMat2dScaleLocal</span><span data-if="python" style="display:none;">hom_mat2d_scale_local</span><span data-if="cpp" style="display:none;">HomMat2dScaleLocal</span><span data-if="c" style="display:none;">hom_mat2d_scale_local</span></code></a></dt>
<dd>Add a scaling to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_slant.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_slant</span><span data-if="dotnet" style="display:none;">HomMat2dSlant</span><span data-if="python" style="display:none;">hom_mat2d_slant</span><span data-if="cpp" style="display:none;">HomMat2dSlant</span><span data-if="c" style="display:none;">hom_mat2d_slant</span></code></a></dt>
<dd>Add a slant to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_slant_local.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_slant_local</span><span data-if="dotnet" style="display:none;">HomMat2dSlantLocal</span><span data-if="python" style="display:none;">hom_mat2d_slant_local</span><span data-if="cpp" style="display:none;">HomMat2dSlantLocal</span><span data-if="c" style="display:none;">hom_mat2d_slant_local</span></code></a></dt>
<dd>Add a slant to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_to_affine_par.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_to_affine_par</span><span data-if="dotnet" style="display:none;">HomMat2dToAffinePar</span><span data-if="python" style="display:none;">hom_mat2d_to_affine_par</span><span data-if="cpp" style="display:none;">HomMat2dToAffinePar</span><span data-if="c" style="display:none;">hom_mat2d_to_affine_par</span></code></a></dt>
<dd>Compute the affine transformation parameters from a homogeneous 2D
transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_translate.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_translate</span><span data-if="dotnet" style="display:none;">HomMat2dTranslate</span><span data-if="python" style="display:none;">hom_mat2d_translate</span><span data-if="cpp" style="display:none;">HomMat2dTranslate</span><span data-if="c" style="display:none;">hom_mat2d_translate</span></code></a></dt>
<dd>Add a translation to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_translate_local.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_translate_local</span><span data-if="dotnet" style="display:none;">HomMat2dTranslateLocal</span><span data-if="python" style="display:none;">hom_mat2d_translate_local</span><span data-if="cpp" style="display:none;">HomMat2dTranslateLocal</span><span data-if="c" style="display:none;">hom_mat2d_translate_local</span></code></a></dt>
<dd>Add a translation to a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat2d_transpose.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat2d_transpose</span><span data-if="dotnet" style="display:none;">HomMat2dTranspose</span><span data-if="python" style="display:none;">hom_mat2d_transpose</span><span data-if="cpp" style="display:none;">HomMat2dTranspose</span><span data-if="c" style="display:none;">hom_mat2d_transpose</span></code></a></dt>
<dd>Transpose a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_mat3d_project.html"><code><span data-if="hdevelop" style="display:inline;">hom_mat3d_project</span><span data-if="dotnet" style="display:none;">HomMat3dProject</span><span data-if="python" style="display:none;">hom_mat3d_project</span><span data-if="cpp" style="display:none;">HomMat3dProject</span><span data-if="c" style="display:none;">hom_mat3d_project</span></code></a></dt>
<dd>Project an affine 3D transformation matrix to a 2D projective
transformation matrix.</dd>
</dl>
<dl>
<dt><a href="hom_vector_to_proj_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">hom_vector_to_proj_hom_mat2d</span><span data-if="dotnet" style="display:none;">HomVectorToProjHomMat2d</span><span data-if="python" style="display:none;">hom_vector_to_proj_hom_mat2d</span><span data-if="cpp" style="display:none;">HomVectorToProjHomMat2d</span><span data-if="c" style="display:none;">hom_vector_to_proj_hom_mat2d</span></code></a></dt>
<dd>Compute a homogeneous transformation matrix using given point
correspondences.</dd>
</dl>
<dl>
<dt><a href="point_line_to_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">point_line_to_hom_mat2d</span><span data-if="dotnet" style="display:none;">PointLineToHomMat2d</span><span data-if="python" style="display:none;">point_line_to_hom_mat2d</span><span data-if="cpp" style="display:none;">PointLineToHomMat2d</span><span data-if="c" style="display:none;">point_line_to_hom_mat2d</span></code></a></dt>
<dd>Approximate an affine transformation from point-to-line correspondences.</dd>
</dl>
<dl>
<dt><a href="projective_trans_pixel.html"><code><span data-if="hdevelop" style="display:inline;">projective_trans_pixel</span><span data-if="dotnet" style="display:none;">ProjectiveTransPixel</span><span data-if="python" style="display:none;">projective_trans_pixel</span><span data-if="cpp" style="display:none;">ProjectiveTransPixel</span><span data-if="c" style="display:none;">projective_trans_pixel</span></code></a></dt>
<dd>Project pixel coordinates using a homogeneous projective
transformation matrix.</dd>
</dl>
<dl>
<dt><a href="projective_trans_point_2d.html"><code><span data-if="hdevelop" style="display:inline;">projective_trans_point_2d</span><span data-if="dotnet" style="display:none;">ProjectiveTransPoint2d</span><span data-if="python" style="display:none;">projective_trans_point_2d</span><span data-if="cpp" style="display:none;">ProjectiveTransPoint2d</span><span data-if="c" style="display:none;">projective_trans_point_2d</span></code></a></dt>
<dd>Project a homogeneous 2D point using a projective transformation
matrix.</dd>
</dl>
<dl>
<dt><a href="serialize_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">serialize_hom_mat2d</span><span data-if="dotnet" style="display:none;">SerializeHomMat2d</span><span data-if="python" style="display:none;">serialize_hom_mat2d</span><span data-if="cpp" style="display:none;">SerializeHomMat2d</span><span data-if="c" style="display:none;">serialize_hom_mat2d</span></code></a></dt>
<dd>Serialize a homogeneous 2D transformation matrix.</dd>
</dl>
<dl>
<dt><a href="vector_angle_to_rigid.html"><code><span data-if="hdevelop" style="display:inline;">vector_angle_to_rigid</span><span data-if="dotnet" style="display:none;">VectorAngleToRigid</span><span data-if="python" style="display:none;">vector_angle_to_rigid</span><span data-if="cpp" style="display:none;">VectorAngleToRigid</span><span data-if="c" style="display:none;">vector_angle_to_rigid</span></code></a></dt>
<dd>Compute a rigid affine transformation from points and angles.</dd>
</dl>
<dl>
<dt><a href="vector_field_to_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">vector_field_to_hom_mat2d</span><span data-if="dotnet" style="display:none;">VectorFieldToHomMat2d</span><span data-if="python" style="display:none;">vector_field_to_hom_mat2d</span><span data-if="cpp" style="display:none;">VectorFieldToHomMat2d</span><span data-if="c" style="display:none;">vector_field_to_hom_mat2d</span></code></a></dt>
<dd>Approximate an affine map from a displacement vector field.</dd>
</dl>
<dl>
<dt><a href="vector_to_aniso.html"><code><span data-if="hdevelop" style="display:inline;">vector_to_aniso</span><span data-if="dotnet" style="display:none;">VectorToAniso</span><span data-if="python" style="display:none;">vector_to_aniso</span><span data-if="cpp" style="display:none;">VectorToAniso</span><span data-if="c" style="display:none;">vector_to_aniso</span></code></a></dt>
<dd>Approximate an anisotropic similarity transformation from point
correspondences.</dd>
</dl>
<dl>
<dt><a href="vector_to_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">vector_to_hom_mat2d</span><span data-if="dotnet" style="display:none;">VectorToHomMat2d</span><span data-if="python" style="display:none;">vector_to_hom_mat2d</span><span data-if="cpp" style="display:none;">VectorToHomMat2d</span><span data-if="c" style="display:none;">vector_to_hom_mat2d</span></code></a></dt>
<dd>Approximate an affine transformation from point correspondences.</dd>
</dl>
<dl>
<dt><a href="vector_to_proj_hom_mat2d.html"><code><span data-if="hdevelop" style="display:inline;">vector_to_proj_hom_mat2d</span><span data-if="dotnet" style="display:none;">VectorToProjHomMat2d</span><span data-if="python" style="display:none;">vector_to_proj_hom_mat2d</span><span data-if="cpp" style="display:none;">VectorToProjHomMat2d</span><span data-if="c" style="display:none;">vector_to_proj_hom_mat2d</span></code></a></dt>
<dd>Compute a projective transformation matrix using given point
correspondences.</dd>
</dl>
<dl>
<dt><a href="vector_to_proj_hom_mat2d_distortion.html"><code><span data-if="hdevelop" style="display:inline;">vector_to_proj_hom_mat2d_distortion</span><span data-if="dotnet" style="display:none;">VectorToProjHomMat2dDistortion</span><span data-if="python" style="display:none;">vector_to_proj_hom_mat2d_distortion</span><span data-if="cpp" style="display:none;">VectorToProjHomMat2dDistortion</span><span data-if="c" style="display:none;">vector_to_proj_hom_mat2d_distortion</span></code></a></dt>
<dd>Compute a projective transformation matrix and the radial distortion
coefficient using given image point correspondences.</dd>
</dl>
<dl>
<dt><a href="vector_to_rigid.html"><code><span data-if="hdevelop" style="display:inline;">vector_to_rigid</span><span data-if="dotnet" style="display:none;">VectorToRigid</span><span data-if="python" style="display:none;">vector_to_rigid</span><span data-if="cpp" style="display:none;">VectorToRigid</span><span data-if="c" style="display:none;">vector_to_rigid</span></code></a></dt>
<dd>Approximate a rigid affine transformation from point correspondences.</dd>
</dl>
<dl>
<dt><a href="vector_to_similarity.html"><code><span data-if="hdevelop" style="display:inline;">vector_to_similarity</span><span data-if="dotnet" style="display:none;">VectorToSimilarity</span><span data-if="python" style="display:none;">vector_to_similarity</span><span data-if="cpp" style="display:none;">VectorToSimilarity</span><span data-if="c" style="display:none;">vector_to_similarity</span></code></a></dt>
<dd>Approximate an similarity transformation from point correspondences.</dd>
</dl>
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